Explanation

(d)
Let $AD$ be the perpendicular and $D$ be the foot of the perpendicular which divides $BC$ in the ratio $\lambda:1$, then
$D\left( \frac{10\lambda - 9}{\lambda + 1},\frac{4}{\lambda + 1},\frac{- \lambda + 5}{\lambda + 1} \right)$
[image:12_image9.emf]
The direction ratios of $AD$ are $\frac{10\lambda - 9}{\lambda + 1},\frac{4}{\lambda + 1}\$and $\frac{–\lambda + 5}{\lambda + 1}$ and direction ratios of $BC$ are $19, - 4$ and $- 6$
Since $AD\bot BC$, we get
$19\left( \frac{10\lambda - 9}{\lambda + 1} \right) - 4\left( \frac{4}{\lambda + 1} \right) - 6\left( \frac{- \lambda + 5}{\lambda + 1} \right) = 0$
$\Rightarrow \ \lambda = \frac{31}{28}$
Hence, on putting the value of $\lambda$ in (i), we get required foot of the perpendicular, i.e., $\left( \frac{58}{59},\frac{112}{59},\frac{109}{59} \right)$